Question

Лыжник массой 85 кг спускается с постоянной скоростью со склона, составляющего 10° с горизонталью. Найти коэффициент трения лыж о снег. С каким ускорением лыжник будет спускаться с 15°-склона при сохранении условий скольжения?

Answer 1

Запишем второй закон Ньютона для первого угла спуска (все ускорения лыжника равны 0, выбираем проекции на оси вдоль плоскости и перпендикулярно ей)

$0 = mg\sin\alpha_1 - \mu N\\ 0 = mg\cos\alpha_1 - N\\\\ N = mg\cos\alpha_1\\ \mu = mg\sin\alpha_1/N = \tan\alpha_1$

Нашли коэффициент трения. Чтобы найти ускорение лыжника со склона покруче, опять-таки напишем второй закон Ньютона с учетом вышеприведенного равенства (ускорение ненулевое только вдоль склона)

$ma = mg\sin\alpha_2-\tan\alpha_1 mg\cos\alpha_2\\ a = g(\sin\alpha_2 - \tan\alpha_1\cos\alpha_2)$

Answer 2

Calculation of the Coefficient of Friction

To find the coefficient of friction between the skis and the snow, we can use the given information about the skier's mass, the slope angle, and the constant speed of descent.

The formula for the coefficient of friction can be derived from the forces acting on the skier. In this case, the only significant force is the force of friction opposing the skier's motion. This force can be expressed as the product of the coefficient of friction and the normal force acting on the skier.

The normal force is equal to the component of the skier's weight perpendicular to the slope. It can be calculated using the formula:

Normal force = mass * gravitational acceleration * cos(slope angle)

where the mass is given as 85 kg, the gravitational acceleration is approximately 9.8 m/s^2, and the slope angle is 10°.

Using this formula, we can calculate the normal force acting on the skier:

Normal force = 85 kg * 9.8 m/s^2 * cos(10°)

Now, we can use the fact that the skier is descending with a constant speed to determine the frictional force. Since the skier is moving at a constant speed, the net force acting on the skier must be zero. Therefore, the frictional force must be equal in magnitude and opposite in direction to the component of the skier's weight parallel to the slope.

The component of the skier's weight parallel to the slope can be calculated using the formula:

Component of weight parallel to slope = mass * gravitational acceleration * sin(slope angle)

Using this formula, we can calculate the component of the skier's weight parallel to the slope:

Component of weight parallel to slope = 85 kg * 9.8 m/s^2 * sin(10°)

Since the frictional force is equal in magnitude to the component of the skier's weight parallel to the slope, we can equate the two expressions:

Frictional force = mass * gravitational acceleration * sin(slope angle)

Now, we can substitute the given values into the equation to find the frictional force:

Frictional force = 85 kg * 9.8 m/s^2 * sin(10°)

To find the coefficient of friction, we divide the frictional force by the normal force:

Coefficient of friction = Frictional force / Normal force

Now, let's calculate the coefficient of friction using the given values:

Coefficient of friction = (85 kg * 9.8 m/s^2 * sin(10°)) / (85 kg * 9.8 m/s^2 * cos(10°))

Calculating this expression gives us the coefficient of friction between the skis and the snow.

Calculation of the Acceleration on a 15° Slope

To find the acceleration of the skier on a 15° slope while maintaining the conditions of sliding, we can use the same approach as before.

The component of the skier's weight parallel to the slope can be calculated using the formula:

Component of weight parallel to slope = mass * gravitational acceleration * sin(slope angle)

Using this formula, we can calculate the component of the skier's weight parallel to the 15° slope:

Component of weight parallel to slope = 85 kg * 9.8 m/s^2 * sin(15°)

Since the skier is sliding, the frictional force must be equal in magnitude to the component of the skier's weight parallel to the slope. Therefore, the frictional force can be calculated as:

Frictional force = mass * gravitational acceleration * sin(slope angle)

Using this formula, we can calculate the frictional force:

Frictional force = 85 kg * 9.8 m/s^2 * sin(15°)

The net force acting on the skier is equal to the difference between the component of the skier's weight parallel to the slope and the frictional force. Since the skier is sliding, the net force must be equal to the product of the skier's mass and acceleration:

Net force = mass * acceleration

Setting the net force equal to the difference between the component of the skier's weight parallel to the slope and the frictional force, we can solve for the acceleration:

mass * acceleration = (mass * gravitational acceleration * sin(slope angle)) - (mass * gravitational acceleration * sin(slope angle))

Simplifying the equation, we find:

acceleration = 0

This means that the skier will not experience any acceleration while sliding down a 15° slope, as long as the conditions of sliding are maintained.

Please note that the calculations provided above are based on the given information and assumptions about the conditions of sliding.